Princeton University Special Physics Seminar

Abstract: Entanglement growth at late times is explained by the membrane picture in various chaotic quantum systems. In this work, we discuss entanglement growth in the large N Brownian SYK chain. We prepare an infinite-temperature thermofield double state of two chains (L and R) and evolve chain R by time T. We consider the growth of Rényi entropies for bipartitions, including all sites to the left of 'x’ on chain L and to the left of 'y' on chain R. The average growth of Rényi-n entropy in the Brownian SYK chain is mapped to Euclidean evolution by a positive Hamiltonian that acts on 2n copies of the system. The initial and final states in the evolution contain a domain wall at site y and site x, respectively, which separates two ground states of the Hamiltonian. The effective dynamics is described by a membrane in spacetime that connects the two domain walls. We show that the membrane has a width that depends on the velocity v =(x-y)/T. The width monotonically increases with the velocity and diverges as -log(vB-v) as v approaches vB (the butterfly velocity). For v > vB, the membrane splits into two fronts, each moving with the butterfly velocity. The two fronts are separated by a new domain of width (v-vB)T. We argue that this new domain is a metastable state that arises in the large N limit.